Briton Froome looked set to win the stage after riding clear of his general classification rivals on a 3.5km ascent that featured gradients of 19%.
However, he was caught in the final 50 metres by Dumoulin, who regained the overall race lead from Esteban Chaves. Team Sky's Froome is now eighth overall, 78 seconds behind Dumoulin.
However, with the Dutchman not expected to be in contention when the 21-stage race reaches the high mountains, of more interest was the time Froome clawed back on his rivals for the overall win.
Colombia's Nairo Quintana, another of the race favourites, finished 18 seconds adrift and Froome now trails the Movistar rider by just one second.
The Tour de France winner also finished ahead of Joaquim Rodriguez and Alejandro Valverde, the 2009 champion, and he is just 21 and nine seconds respectively behind the Spanish pair.
After narrowly losing out to Dumoulin for the stage win, Froome said: "I'm gutted, I gave it everything.
"I thought I had it for a second there. But Dumoulin has shown incredible form in this race. He's a young rider with a bright future ahead of him.
"The guys did a fantastic job for me today keeping myself, Nicolas Roche, Mikel Nieve and Sergio Henao - the four climbers - at the front of the race, and bringing us into the climb in a perfect position. So we've got them to thank for all the hard work they did."
Colombia's Chaves, who went into stage nine with a 10-second lead over Giant's Dumoulin, finally cracked in the final couple of kilometres as first the Dutchman and then Froome attacked.
Dumoulin, who was race leader after five stages, holds a 57-second advantage over Rodriguez, with Chaves third, two seconds further back while Froome's Team Sky team-mate Nicolas Roche of Ireland is fourth, 67 seconds adrift.
Britain's Geraint Thomas spent most of the 168.3km (104.5-mile) race from Torrevieja to Cumbre del Sol, or 'Summit of the Sun', in the break, finally giving up when his group was caught at the bottom of the finishing climb.
Monday's 10th stage is a 146.6km race from Valencia to Castellon.
Stage nine result:
1. Tom Dumoulin (Ned/Giant) 4hrs 09mins 55secs
2. Chris Froome (GB/ Team Sky) +02secs
3. Joaquim Rodriguez (Spa/ Katusha) +05secs
4. Fabio Aru (Ita/Astana) +16secs
5. Rafal Majka (Pol/Tinkoff - Saxo) +18secs
6. Nairo Quintana (Col/Movistar) +20secs
7. Alejandro Valverde (Spa/Movistar) Same time
8. Nicolas Roche (Ire/Team Sky) +31secs
9. Domenico Pozzovivo (Ita/AG2R) +33secs
10. Louis Meintjes (SA/Team MTN) +34secs
Selected:
15. Esteban Chaves (Col/Orica) +59secs
General classification after stage nine:
1. Tom Dumoulin (Ned/Giant) 35hrs 22mins 13secs
2. Joaquim Rodriguez (Spa/Katusha) +57secs
3. Esteban Chaves (Col/Orica) +59secs
4. Nicolas Roche (Ire/Team Sky) +1min 07secs
5. Alejandro Valverde (Spa/Movistar) +1min 09secs
6. Fabio Aru (Ita/Astana) +1min 13secs
7. Nairo Quintana (Col/Movistar) +1min 17secs
8. Chris Froome (GB/Team Sky) +1min 18secs
9. Rafal Majka (Pol/Tinkoff - Saxo) +1min 47secs
10. Domenico Pozzovivo (Ita/AG2R) +1min 52secs
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